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Chapter 9

Chapter 9. Solids and Fluids. Contents. States of Matter Stress vs. Strain Stress vs. Strain curve: One dimensional, two dimensional, and three dimensional objects. Young’s, Shear, and Bulk modulus of elasticity Fracture Arches and Domes Density and Pressure

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Chapter 9

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  1. Chapter 9 Solids and Fluids

  2. Contents • States of Matter • Stress vs. Strain • Stress vs. Strain curve: One dimensional, two dimensional, and three dimensional objects. Young’s, Shear, and Bulk modulus of elasticity • Fracture • Arches and Domes • Density and Pressure • Variation of pressure with depth • Pressure measurements • Bouyant forces and Archimedes principle • Applications and Problems solving

  3. States of Matter • Solid • Liquid • Gas • Plasma

  4. Solids • Have definite volume • Have definite shape • Molecules are held in specific locations • By electrical forces • Vibrate about their equilibrium positions • Can be modeled as springs connecting molecules

  5. More About Solids • External forces can be applied to the solid and compress the material • In the model, the springs would be compressed • When the force is removed, the solid returns to its original shape and size • This property is called elasticity

  6. Crystalline Solid • Atoms have an ordered structure • Long range order • This example is salt • Gray spheres represent Na+ ions • Green spheres represent Cl- ions

  7. Amorphous Solid • Atoms are arranged almost randomly • Examples include glass

  8. Liquid • Has a definite volume • No definite shape • Exists at a higher temperature than solids • The molecules “wander” through the liquid in a random fashion • The intermolecular forces are not strong enough to keep the molecules in a fixed position

  9. Gas • Has no definite volume • Has no definite shape • Molecules are in constant random motion • The molecules exert only weak forces on each other • Average distance between molecules is large compared to the size of the molecules

  10. Plasma • Gas heated to a very high temperature • Many of the electrons are freed from the nucleus • Result is a collection of free, electrically charged ions • Plasmas exist inside stars

  11. Types of Matter • Normal matter • About 5% of total matter in the total universe • Dark matter • Affects the motion of stars in galaxies • May be as much at 25% of total matter • Dark energy • Accounts for acceleration of the expansion of the universe • May be as much as 70% of all matter

  12. Deformation of Solids • All objects are deformable • It is possible to change the shape or size (or both) of an object through the application of external forces • When the forces are removed, the object tends to its original shape • An object undergoing this type of deformation exhibits elastic behavior

  13. Elastic Properties • Stress is the force per unit area causing the deformation • Strain is a measure of the amount of deformation • The elastic modulus is the constant of proportionality between stress and strain • For sufficiently small stresses, the stress is directly proportional to the strain • The constant of proportionality depends on the material being deformed and the nature of the deformation

  14. Elastic Modulus • The elastic modulus can be thought of as the stiffness of the material • A material with a large elastic modulus is very stiff and difficult to deform • Analogous to the spring constant

  15. Young’s Modulus: Elasticity in Length • Tensile stress is the ratio of the external force to the cross-sectional area • Tensile is because the bar is under tension • The elastic modulus is called Young’s modulus

  16. Young’s Modulus, cont. • SI units of stress are Pascals, Pa • 1 Pa = 1 N/m2 • The tensile strain is the ratio of the change in length to the original length • Strain is dimensionless

  17. Young’s Modulus Ultimate strength • Young’s modulus applies to a stress of either tension or compression • It is possible to exceed the elastic limit of the material • No longer directly proportional • Ordinarily does not return to its original length

  18. Types of stress for rigid bodies

  19. Breaking • If stress continues, it surpasses its ultimate strength • The ultimate strength is the greatest stress the object can withstand without breaking • The breaking point • For a brittle material, the breaking point is just beyond its ultimate strength • For a ductile material, after passing the ultimate strength the material thins and stretches at a lower stress level before breaking

  20. Shear Modulus:Elasticity of Shape • Forces may be parallel to one of the object’s faces • The stress is called a shear stress • The shear strain is the ratio of the horizontal displacement and the height of the object • The shear modulus is S

  21. Shear Modulus, Equations • S is the shear modulus • A material having a large shear modulus is difficult to bend

  22. Shear Modulus, final • There is no volume change in this type of deformation • Remember the force is parallel to the cross-sectional area • In tensile stress, the force is perpendicular to the cross-sectional area

  23. Bulk Modulus:Volume Elasticity • Bulk modulus characterizes the response of an object to uniform squeezing • Suppose the forces are perpendicular to, and act on, all the surfaces • Example: when an object is immersed in a fluid • The object undergoes a change in volume without a change in shape

  24. Bulk Modulus, cont. • Volume stress, ΔP, is the ratio of the force to the surface area • This is also called the Pressure when dealing with fluids • The volume strain is equal to the ratio of the change in volume to the original volume

  25. Bulk Modulus, final • A material with a large bulk modulus is difficult to compress • The negative sign is included since an increase in pressure will produce a decrease in volume • B is always positive • The compressibility is the reciprocal of the bulk modulus

  26. Stress vs. Strain Relationship • F/A = Y (L/L) : Y = Young’s Modulus (E) • F/A = S (x/h) : S = Shear Modulus (G) • (F/A) = -B (V/V), or P = -B V/V, B = Bulk modulus (notice the negative sign). • Consult table 9.1 for Elastic modulii.

  27. Notes on Moduli • Solids have Young’s, Bulk, and Shear moduli • Liquids have only bulk moduli, they will not undergo a shearing or tensile stress • The liquid would flow instead

  28. Post and Beam Arches • A horizontal beam is supported by two columns • Used in Greek temples • Columns are closely spaced • Limited length of available stones • Low ultimate tensile strength of sagging stone beams

  29. Semicircular Arch • Developed by the Romans • Allows a wide roof span on narrow supporting columns • Stability depends upon the compression of the wedge-shaped stones

  30. Gothic Arch • First used in Europe in the 12th century • Extremely high • The flying buttresses are needed to prevent the spreading of the arch supported by the tall, narrow columns

  31. Density • The density of a substance of uniform composition is defined as its mass per unit volume: • Units are kg/m3 (SI) or g/cm3 (cgs) • 1 g/cm3 = 1000 kg/m3

  32. Density, cont. • The densities of most liquids and solids vary slightly with changes in temperature and pressure • Densities of gases vary greatly with changes in temperature and pressure • The higher normal densities of solids and liquids compared to gases implies that the average spacing between molecules in a gas is about 10 times greater than the solid or liquid

  33. Specific Gravity • The specific gravity of a substance is the ratio of its density to the density of water at 4° C • The density of water at 4° C is 1000 kg/m3 • Specific gravity is a unitless ratio

  34. Pressure • The force exerted by a fluid on a submerged object at any point is perpendicular to the surface of the object

  35. Variation of Pressure with Depth • If a fluid is at rest in a container, all portions of the fluid must be in static equilibrium • All points at the same depth must be at the same pressure • Otherwise, the fluid would not be in equilibrium • The fluid would flow from the higher pressure region to the lower pressure region

  36. Pressure and Depth • Examine the darker region, assumed to be a fluid • It has a cross-sectional area A • Extends to a depth h below the surface • Three external forces act on the region

  37. Pressure and Depth equation • Po is normal atmospheric pressure • 1.013 x 105 Pa = 14.7 lb/in.2 • The pressure does not depend upon the shape of the container

  38. Pascal’s Principle • A change in pressure applied to an enclosed fluid is transmitted undimished to every point of the fluid and to the walls of the container. • First recognized by Blaise Pascal, a French scientist (1623 – 1662)

  39. Pascal’s Principle, cont • The hydraulic press is an important application of Pascal’s Principle • Also used in hydraulic brakes, forklifts, car lifts, etc.

  40. Absolute vs. Gauge Pressure • The pressure P is called the absolute pressure • Remember, P = Po + rgh • P – Po = rgh is the gauge pressure

  41. Pressure Measurements:Manometer • One end of the U-shaped tube is open to the atmosphere • The other end is connected to the pressure to be measured • If P in the system is greater than atmospheric pressure, h is positive • If less, then h is negative

  42. Pressure Measurements: Barometer • Invented by Torricelli (1608 – 1647) • A long closed tube is filled with mercury and inverted in a dish of mercury • Measures atmospheric pressure as ρgh

  43. Blood Pressure • Blood pressure is measured with a special type of manometer called a sphygmomano-meter • Pressure is measured in mm of mercury

  44. Pressure Values in Various Units • One atmosphere of pressure is defined as the pressure equivalent to a column of mercury exactly 0.76 m tall at 0o C where g = 9.806 65 m/s2 • One atmosphere (1 atm) = • 76.0 cm of mercury • 1.013 x 105 Pa • 14.7 lb/in2

  45. Archimedes • 287 – 212 BC • Greek mathematician, physicist, and engineer • Buoyant force • Inventor

  46. Archimedes' Principle • Any object completely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the weight of the fluid displaced by the object

  47. Buoyant Force • The upward force is called the buoyant force • The physical cause of the buoyant force is the pressure difference between the top and the bottom of the object

  48. Buoyant Force, cont. • The magnitude of the buoyant force always equals the weight of the displaced fluid • The buoyant force is the same for a totally submerged object of any size, shape, or density

  49. Buoyant Force, final • The buoyant force is exerted by the fluid • Whether an object sinks or floats depends on the relationship between the buoyant force and the weight

  50. Archimedes’ Principle:Totally Submerged Object • The upward buoyant force is B=ρfluidgVobj • The downward gravitational force is w=mg=ρobjgVobj • The net force is B-w=(ρfluid-ρobj)gVobj

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